Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting

The simple, little and slow things count : on parameterized counting complexity

In this thesis, we study the parameterized complexity of counting problems, as introduced by Flum and Grohe. This area mainly involves questions of the following kind: On inputs x with a parameter k, can we solve a given counting problem in time f(k)*|x|^c for a function f that depends only on k? In the positive case, we call the problem fixed-parameter tractable (fpt). Otherwise, we try to prove its #W[1]-hardness, which is the parameterized analogue of #P-hardness. We introduce a general technique that bridges parameterized counting complexity and the so-called Holant framework. We then apply this technique to the problem of counting perfect matchings (or equivalently, the permanent) subject to structural parameters of the input graph G: On the algorithmic side, we introduce a new tractable structural parameter, namely, the minimal size of an excluded single-crossing minor of G. We complement this by showing that counting perfect matchings is #W[1]-hard when parameterized by the size of an arbitrary excluded minor. Then we turn our attention to counting general subgraphs H other than perfect matchings in a host graph G. Instead of imposing structural parameters on G, we parameterize by the size of H, giving rise to the problems #Sub(C) for fixed graph classes C: For inputs H and G with H in C, we wish to count H-copies in G. Here, C could be the class of matchings, cycles, paths, or any other recursively enumerable class. We give a full dichotomy for these problems: Either #Sub(C) has a polynomial-time algorithm or it is #W[1]-complete. Assuming that FPT and #W[1] do not coincide, we can thus precisely identify the graph classes C for which the subgraph counting problem #Sub(C) admits polynomial-time algorithms. Furthermore, we obtain an unexpected application of our extensions to the Holant framework: We show that, given two unweighted graphs, it is C=P-complete to decide whether they have the same number of perfect matchings. Finally, we prove conditional lower bounds for counting problems under the counting exponential-time hypothesis #ETH. This hypothesis, introduced by Dell et al., asserts that the satisfying assignments to n-variable formulas in 3-CNF cannot be counted in time 2^o(n). Building upon this, we introduce a general technique that allows to derive tight lower bounds for other counting problems, such as counting perfect matchings, the Tutte polynomial, and the matching polynomial.Die vorliegende Arbeit befasst sich mit der parametrisierten Komplexität von Zählproblemen, einem von Flum und Grohe gegründeten Gebiet, in welchem Fragen der folgenden Art betrachtet werden: Können gegebene Probleme auf Eingaben x mit Parameter k in Zeit f(k)*|x|^c gelöst werden, wobei f eine Funktion ist, die nur von k abhängt? Im positiven Falle bezeichnen wir das Problem als parametrisierbar (FPT). Andernfalls versuchen wir typischerweise, dessen #W[1]-Härte zu beweisen - diese lässt sich vereinfachend als ein parametrisiertes Äquivalent der #P-Härte auffassen. Wir führen zunächst eine allgemeine Technik ein, welche die parametrisierte Zählkomplexität mit dem sogenannten Holant-Rahmenwerk verbindet. Anschließend setzen wir diese zum Zählen perfekter Paarungen (oder äquivalent, zur Auswertung der Permanente) unter strukturellen Parametern des Eingabegraphens G ein: Wir zeigen, dass das Zählen perfekter Paarungen parametrisierbar ist durch die minimale Größe eines ausgeschlossenen Minors von G, der höchstens eine Kreuzung besitzt. Dieses algorithmische Resultat komplementieren wir durch die #W[1]-Härte des Zählens perfekter Paarungen, wenn die minimale Größe eines beliebigen ausgeschlossenen Minors als Parameter betrachtet wird. Anschließend widmen wir uns dem Zählen beliebiger Subgraphen H in Graphen G. Anstelle von strukturellen Parametern betrachten wir die Größe von H als Parameter und erhalten hierdurch die Probleme #Sub(C) für feste Graphklassen C: Auf Eingaben H und G mit H in C gilt es, die H-Kopien in G zu zählen. Hierbei kann C die Klasse der Paarungen, Zyklen, Pfade, oder eine beliebige andere Klasse von Graphen darstellen. Wir zeigen eine vollständige Dichotomie für diese Probleme: Das Problem #Sub(C) ist entweder in P oder #W[1]-hart. Unter der gängigen Annahme, dass FPT und #W[1] nicht zusammenfallen, erhalten wir somit eine vollständige Klassifikation der Polynomialzeit-lösbaren Probleme #Sub(C). Weiterhin erhalten wir eine unerwartete Anwendung unserer Erweiterungen des Holant-Rahmenwerks: Wir zeigen die C=P-Vollständigkeit der Frage, ob die Anzahlen perfekter Paarungen in zwei gegebenen ungewichteten Graphen übereinstimmen. Schlussendlich zeigen wir bedingte untere Schranken für Zählprobleme unter der Zählversion der Exponentialzeithypothese #ETH, eingeführt durch Dell et al. Diese postuliert, dass die erfüllenden Belegungen in 3-KNF-Formeln mit n Variablen nicht in Zeit 2^o(n) gezählt werden können. Darauf aufbauend führen wir eine allgemeine Technik ein, die es ermöglicht, scharfe untere Schranken für andere Zählprobleme zu erhalten: Dies umfasst das Zählen perfekter Paarungen, das Tutte-Polynom und das Paarungs-Polynom

Approximate counting via complex zero-free regions and spectral independence

This thesis investigates fundamental problems in approximate counting that arise in the field of statistical mechanics. Building upon recent advancements in the area, our research aims to enhance our understanding of the computational complexity of sampling from the Ising and Potts models, as well as the random $k$-SAT model. The $q$-state Potts model is a spin model in which each particle is randomly assigned a spin (out of $q$ possible spins), where the probability of a certain assignment depends on how many adjacent particles present the same spin. The edge interaction of the model is a parameter that quantifies the strength of interaction between two adjacent particles. The Ising model corresponds to the Potts model with $q = 2$. Sampling from these models is inherently connected to approximating the partition function of the model, a graph polynomial that encodes several aggregate thermodynamic properties of the system. In addition to classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane of these partition functions, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Thus, following this trend in both statistical physics and algorithmic research, we allow the edge interaction to be any complex number. First, we study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations. Secondly, we investigate the complexity of approximating the partition function \ising(G; \beta) of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. In this thesis we establish both new tractability and inapproximability results. Our tractability results show that \ising(-; \beta) has an FPTAS when $\beta \in \mathbb{C}$ and $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating \ising(-, \beta) on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model. Finally, we exploit the recently successful framework of spectral independence to analyse the mixing time of a Markov chain, and we apply it in order to sample satisfying assignments of $k$-CNF formulas. Our analysis leads to a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the random formula $\alpha=m/n$ scales exponentially with $k$, where $n$ is the number of variables and $m$ is the number of clauses. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $\alpha=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(\Theta(k))}$. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $\alpha\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting

Counting Problems on Quantum Graphs: Parameterized and Exact Complexity Classifications

Quantum graphs, as defined by Lovász in the late 60s, are formal linear combinations of simple graphs with finite support. They allow for the complexity analysis of the problem of computing finite linear combinations of homomorphism counts, the latter of which constitute the foundation of the structural hardness theory for parameterized counting problems: The framework of parameterized counting complexity was introduced by Flum and Grohe, and McCartin in 2002 and forms a hybrid between the classical field of computational counting as founded by Valiant in the late 70s and the paradigm of parameterized complexity theory due to Downey and Fellows which originated in the early 90s. The problem of computing homomorphism numbers of quantum graphs subsumes general motif counting problems and the complexity theoretic implications have only turned out recently in a breakthrough regarding the parameterized subgraph counting problem by Curticapean, Dell and Marx in 2017. We study the problems of counting partially injective and edge-injective homomorphisms, counting induced subgraphs, as well as counting answers to existential first-order queries. We establish novel combinatorial, algebraic and even topological properties of quantum graphs that allow us to provide exhaustive parameterized and exact complexity classifications, including necessary, sufficient and mostly explicit tractability criteria, for all of the previous problems.Diese Arbeit befasst sich mit der Komplexit atsanalyse von mathematischen Problemen die als Linearkombinationen von Graphhomomorphismenzahlen darstellbar sind. Dazu wird sich sogenannter Quantengraphen bedient, bei denen es sich um formale Linearkombinationen von Graphen handelt und welche von Lov asz Ende der 60er eingef uhrt wurden. Die Bestimmung der Komplexit at solcher Probleme erfolgt unter dem von Flum, Grohe und McCartin im Jahre 2002 vorgestellten Paradigma der parametrisierten Z ahlkomplexit atstheorie, die als Hybrid der von Valiant Ende der 70er begr undeten klassischen Z ahlkomplexit atstheorie und der von Downey und Fellows Anfang der 90er eingef uhrten parametrisierten Analyse zu verstehen ist. Die Berechnung von Homomorphismenzahlen zwischen Quantengraphen und Graphen subsumiert im weitesten Sinne all jene Probleme, die das Z ahlen von kleinen Mustern in gro en Strukturen erfordern. Aufbauend auf dem daraus resultierenden Durchbruch von Curticapean, Dell und Marx, das Subgraphz ahlproblem betre end, behandelt diese Arbeit die Analyse der Probleme des Z ahlens von partiell- und kanteninjektiven Homomorphismen, induzierten Subgraphen, und Tre ern von relationalen Datenbankabfragen die sich als existentielle Formeln ausdr ucken lassen. Insbesondere werden dabei neue kombinatorische, algebraische und topologische Eigenschaften von Quantengraphen etabliert, die hinreichende, notwendige und meist explizite Kriterien f ur die Existenz e zienter Algorithmen liefern

The Complexity of Homomorphism Reconstructibility

Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where one would like to answer queries or classify graphs solely based on the representation of a graph $G$ as a finite vector of homomorphism counts from some fixed finite set of graphs to $G$. We study the computational complexity of the arguably most fundamental computational problem associated to these representations, the homomorphism reconstructability problem: given a finite sequence of graphs and a corresponding vector of natural numbers, decide whether there exists a graph $G$ that realises the given vector as the homomorphism counts from the given graphs. We show that this problem yields a natural example of an \mathsf{NP}^{#\mathsf{P}}-hard problem, which still can be $\mathsf{NP}$-hard when restricted to a fixed number of input graphs of bounded treewidth and a fixed input vector of natural numbers, or alternatively, when restricted to a finite input set of graphs. We further show that, when restricted to a finite input set of graphs and given an upper bound on the order of the graph $G$ as additional input, the problem cannot be $\mathsf{NP}$-hard unless $\mathsf{P} = \mathsf{NP}$. For this regime, we obtain partial positive results. We also investigate the problem's parameterised complexity and provide fpt-algorithms for the case that a single graph is given and that multiple graphs of the same order with subgraph instead of homomorphism counts are given